A378260 G.f. satisfies A(x) = A(x^2)/M(x), where M(x) = Sum_{n>=1} mu(n)*x^n and mu(n) = A008683(n), the Moebius function of n.

1, 1, 3, 4, 11, 15, 33, 50, 104, 161, 309, 500, 929, 1529, 2757, 4620, 8207, 13874, 24353, 41478, 72327, 123687, 214685, 368232, 637430, 1095201, 1892492, 3255372, 5619323, 9672701, 16685587, 28734098, 49547095, 85347087, 147130261, 253480414, 436911525, 752798677, 1297444411, 2235633198

Offset

1,3

Links

Paul D. Hanna, Table of n, a(n) for n = 1..3000

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

(1) A(x) = A(x^2)/M(x), where M(x) = Sum_{n>=1} mu(n)*x^n.

(2) x = Sum_{n>=1} A(x^(2*n)) / A(x^n).

a(n) ~ c * d^n, where d = 1.723262561763844024160437963573163520188527015264827413326383054228438457576... and c = 0.7859046910881843332272010625259660209978142303560254864659049088867251443... - Vaclav Kotesovec, Nov 30 2024

Example

G.f. A(x) = x + x^2 + 3*x^3 + 4*x^4 + 11*x^5 + 15*x^6 + 33*x^7 + 50*x^8 + 104*x^9 + 161*x^10 + 309*x^11 + 500*x^12 + 929*x^13 + 1529*x^14 + 2757*x^15 + 4620*x^16 + ...
where A(x) = A(x^2)/M(x) with
M(x) = x - x^2 - x^3 - x^5 + x^6 - x^7 + x^10 - x^11 - x^13 + x^14 + x^15 - x^17 - x^19 + x^21 + x^22 - x^23 + x^26 - x^29 - x^30 + ... + mu(n)*x^n + ...
so that x = M(x) + M(x^2) + M(x^3) + M(x^4) + ... + M(x^n) + ...
Thus, because M(x) = A(x^2)/A(x), we have
x = A(x^2)/A(x) + A(x^4)/A(x^2) + A(x^6)/A(x^3) + A(x^8)/A(x^4) + A(x^10)/A(x^5) + A(x^12)/A(x^6) + ... + A(x^(2*n))/A(x^n) + ...
SPECIFIC VALUES.
A(t) = 1000 at t = 0.57983979082390078033201288097053684588681918658...
A(t) = 100 at t = 0.575850800621842491687274688724496083876096493693...
A(t) = 10 at t = 0.5429296775693301210019293351373468274776922745760...
A(t) = 9 at t = 0.53946231343810887800940222774498269502147986174360...
A(t) = 8 at t = 0.53525852440539581430297764508815311813586247192451...
A(t) = 7 at t = 0.53004645173922704662750351997680689150327151199058...
A(t) = 6 at t = 0.52339661111093477495939037490084005628700411644551...
A(t) = 5 at t = 0.51458419720941955692565375903201066787604036604586...
A(t) = 4 at t = 0.50227142127888616541434068019839636042944372636880...
A(t) = 3 at t = 0.48364898724179834772275350279540495722010623952244...
A(t) = 2 at t = 0.45148154417138074188660255689175385165406842883889...
A(t) = 1 at t = 0.37847838037693933849966786108068785599206753365459...
A(1/2) = 3.85113240762543882840278502418639089248043784485031...
  where A(1/2) = A(1/4)/M(1/2)
  with M(1/2) = 0.10201133481781036474303639393182435154361049251029...
A(1/3) = 0.70553754549458547877689262864744328280095059724850...
  where A(1/3) = A(1/9)/M(1/3)
  with M(1/3) = 0.18199538670263388782780010030056557322634498013538...
A(1/4) = 0.39285915746199878617465323026428187937371048080708...
  where A(1/4) = A(1/16)/M(1/4)
  with M(1/4) = 0.17108224791836356794497287128799432329181231331328...
A(1/5) = 0.27550965922396685715103103981428480321441405929553...
  where A(1/5) = A(1/25)/M(1/5)
  with M(1/5) = 0.15173128129604728456076208173747135942418710339130...
A(1/6) = 0.13414853338170816574291660065981488877610508998414...
A(1/9) = 0.12840457842551423371933936516424287719901492174905...
A(1/16) = 0.06721122777391310699668932733909384687045264984777...
A(1/25) = 0.04180343360348984362058625595257513524070610827394...

Prog

(PARI) {a(n) = my(A=x, M = sum(m=1, n, moebius(m)*x^m) +x*O(x^n));
for(i=1, #binary(n), A = subst(A, x, x^2)/M ); polcoef(A, n)}
for(n=1, 40, print1(a(n), ", "))

Crossrefs

Cf. A073776, A008683.

Sequence in context: A085368 A041405 A042483 * A002530 A042709 A231067

Adjacent sequences: A378254 A378255 A378256 * A378261 A378263 A378265

Keyword

nonn,new

Author

Paul D. Hanna, Nov 25 2024

Status

approved